Question 56
For an integer $n \ge 2$, the tuple $(1, 2, \dots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n \ge 2$ for which all permutations of $\{1, 2, \dots, n\}$ can appear on the blackboard in this way.